Quantum decay rates in chaotic scattering S. Nonnenmacher (Saclay) + M. Zworski (Berkeley) National AMS Meeting, New Orleans, January 2007 A resonant state for the partially open stadium billiard, computed by C.Schmit.
Outline Classical and quantum scattering, resonances. Chaotic régime: fractal trapped set Semiclassical limit: fractal Weyl law, eigenstates distribution spectral gap for thin trapped sets; link with the topological pressure Proof of the gap for a model with absorbing potential: open covers of the trapped set to compute the pressure a quantum partition of unity decomposing the propagator up to logarithmic times a hyperbolic estimate for the dominant terms
Scattering in Hamiltonian systems V(x,y) x y Hamiltonian scattering system Left: hard obstacles in R 2 Right: Hamiltonian H = ξ2 2 + V (x), with potential V (x) decaying at infinity The flow at energies E > 0 is unbounded, and the quantum Hamiltonian H h = h2 D 2 (resp. H h = h2 2 + V (x)) has a purely continuous spectrum on R +.
Quantum resonances 0 E γ h z j The resolvent (z H h ) 1 may be continued meromorphically from {Im z > 0} to {Im z < 0}. Its poles {z j (h)} are the resonances of H h. Each z j (h) is associated with a metastable (non-normalizable) state of H h, with lifetime τ j = h(2 Im z j ) 1 = long-living state if Im z j = O(h). Questions in the semiclassical régime: Distribution of long-living resonances z j (h) with Re z j E, Im z j = O(h). Description of the metastable states ψ h associated with these resonances.
Distribution of resonances - Trapped set Main semiclassical idea: the semiclassical distribution of resonances near E depends on the set K E = Γ + E Γ E of trapped trajectories at energy E. We always assume that K E is contained in a bounded interaction region. V(x,y) x 0 0 Ex. 1: trapped set of positive volume. Resonances are very close from being real (Im z j = O(h )), and from eigenvalues of an associated closed system. y 0 E γh Im z~ c/h e
Ex. 2: empty trapped set for a convex body: resonances are far from the real axis, Im z j h 2 3, and satisfy a Weyl law, h n+1 [Sjöstrand-Z, 99]. Ex 3: the trapped set = a single unstable orbit [Ikawa 83,Gérard,Sjöstrand..] The resonances form a quasi-lattice (Bohr-Sommerfeld + inverted harm. osc.). 0 E γh Im z< h λ/2
Chaotic scattering We will consider chaotic systems, for which K E is a hyperbolic repeller, with a fractal geometry. Ex. 4: n 3 convex obstacles in R 2 K E = Γ + E Γ E has a fractal (Hausdorff/box) dimension dim(k E ) = 2µ E + 1 (µ E < 1).
Fractal Weyl law Theorem. [Sjöstrand-Z 05] given by a fractal Weyl law In the limit h 0, the number of resonances near E is # { z j E < γ h} C E,γ h µ E, h 0. A bold conjecture would say that should be replaced by. An asymptotic relation of that kind was proven for a special open quantum map [Nonnenmacher-Z 05].
Distribution of resonant states The metastable states associated with long-living resonances have specific phase space distributions: Theorem. [Nonnenmacher-Rubin 05, Nonnenmacher-Z 06] Consider a family of resonant states (ψ h ) h 0 s.t. Re z(h) = E + o(1), Im z(h) = O(h) and ψ h L 2 (inter) = 1. Suppose a semiclassical measure µ is associated with (ψ h ): f C c (T R d ), χ π supp f = 1, χψ h, Op h (f)χψ h h 0 f(ρ) dµ(ρ). Then supp µ Γ + E, and for some λ 0 we have Im z(h) h h 0 λ/2 and L XH µ = λµ.
Ex: resonant state for a single hyperbolic point 2 1 0 1 Phase space picture of the Eckart barrier 2 6 4 2 0 2 4 6 Real and imaginary parts of the first resonant state Density plot of the FBI transform of the first resonant state Top: the phase portrait for p(x, ξ) = ξ 2 + cosh 2 (x), with Γ ± 1 highlighted. Middle: the first resonant state, h = 1/16. Bottom: squared modulus of its FBI tranform. The resonant state was computed by D. Bindel, the FBI transform was provided by L. Demanet. As predicted in the above Theorem, the mass of the FBI transform is concentrated on Γ + 1, with an exponential mass growth in the outgoing direction.
Resonance gap for filamentary repellers [Ikawa 88, Burq 93]: if n convex obstacles in R d are far enough from e.o. gap in the semiclassical ( high-energy) resonance spectrum: g > 0, for h small enough, any resonance with Re z j 1/2 satisfies Im z j g h. Equivalently, the quantum lifetimes τ j (2g) 1. 0 gh E γh [Gaspard-Rice 89] (3 disks in R 2 ): If the dimension µ E < 1/2, which is equivalent with P E (1/2) < 0, then there is a gap, and one can take g = P E (1/2). Here P E (1/2) is the topological pressure. Their argument assumes that, in the semiclassical limit, resonances z j = (hk j ) 2 /2 are approximately given by zeros of the Gutzwiller-Voros ( Selberg) zeta function: Z(k) = ) (1 e ikl ω (1/2+j)λ ω converges abs. for Im k > P (1/2). ω j 0
Scattering on convex co-compact quotients The geodesic flow in the infinite-volume manifold X = Γ\H n+1 is uniformly hyperbolic (κ = 1). For any E > 0 the trapped set K E has dimension 2δ + 1, where δ is the dim. of the limit set Λ(Γ). The resonances z = s(n s) = n2 4 + k2 of X are given by the zeros of Z Selberg (s). [Patterson 76, Sullivan 79, Patterson-Perry 01]: All the zeros are in the halfplane Re s δ Im k δ n/2 (their density is bounded by r δ [Guillopé-Lin-Z 04]). Here we also have δ n/2 = P (1/2) the topological pressure of the geodesic flow. Therefore, if P (1/2) < 0 there is a resonance gap g = P (1/2). Actually, in the semiclassical limit Re k, the gap is P (1/2) +ε, due to phase cancellations [Naud 05].
Resonance gap in Euclidean potential scattering We consider Euclidean scattering by a potential V Cc (R d ). We assume that near some noncritical E > 0 the Hamiltonian flow Φ t is uniformly hyperbolic on K E. E ρ + E ρ ρ Σ t Φ (ρ) Let P E (s) be the topological pressure of the flow Φ t on K E, associated with the unstable Jacobian J t + (ρ) 1 = det ( ) 1. dφ t E ρ + Theorem. [Nonnenmacher-Z 06] Assume the topological pressure P E (1/2) < 0, and take any 0 < g < P E (1/2). Then, for h small enough, the resonances of H h such that Re z j = E + o(1) satisfy Im z j gh. In dimension d = 2, the gap condition P E (1/2) < 0 is equivalent with µ E < 1/2, so we recover the criterium of [Gaspard-Rice] for a smooth flow.
A simpler model: scattering with absorbing potential To avoid the trouble of complex dilation, it is convenient to add a complex potential ia(x) to the quantum Hamiltonian, obtaining the nonselfadjoint operator H h,a = h2 2 + V (x) ia(x). A(x)>1 This potential vanishes in the interaction region. Its role is to absorb (kill) the outgoing wavepackets. A(x)=0 As a result, the spectrum of H h,a near the real axis is made of L 2 eigenvalues instead of resonances. These long-living eigenvalues are expected to behave like the resonances of H h. 0 E γ h A( ) 8
An alternative definition of the topological pressure V 1 Σ V 2 K E V 3 V b To define the pressure of Φ t on K E, one starts from an open cover (V b ) b B of K E. This cover is then refined T times through the flow, producing sets V b = V b0 Φ 1 V b1 Φ 2 V b2 Φ T +1 V bt 1, b B T. Keep the V b intersecting K E.
Weigh each V b using the coarse-grained unstable Jacobian: w T (V b ) def = sup ρ V b K E ( J + T (ρ)) 1/2 e T (d 1) λ/2, where λ is an average stretching exponent for initial points in V b. One then considers the partition function Z T def = inf{ The pressure P E (1/2) is finally given by b BT w T (V b ) : B T B T, K E P E (1/2) = lim diam(v b ) 0 lim T 1 T log Z T. b BT V b }. For a given ɛ > 0, we may select a (fine) partition (V b ), a time t 0 N and a cover (V b ) b Bt0 def = (W a ) a A1, such that that { ( w t0 (W a ) exp t0 PE (1/2) + ɛ )}. a A 1
Completing the cover Σ Σ W a U Σ Γ ρ + ρ Γ ρ + ρ Γ ρ We need to complete (W a ) a A1 in order to cover a thin energy layer H 1 ([E δ, E +δ]). One set lies in the forbidden zone: W = {A(x) > 1}. Each of the remaining sets (W a ) a A2 must have escaped to W at the time N 0 t 0 or N 0 t 0, for some N 0 N. We thus get a cover H 1 ([E δ, E + δ]) W a. Γ ρ a A 1 A 2
A quantum partition of unity To this open cover we associate a smooth partition of unity: 1 = π a + π + π E, a A 1 A 2 where π a Cc (W a ), π is supported in W, and π E vanishes in H 1 ([E ± δ/2]). This partition may be h-quantized into a sum of bounded ΨDOs: Id L 2 = Π a, A tot = A 1 A 2 E, Π a = Op w h (π a ). a A tot We use this quantum partition to split the absorbing propagator U = e it 0H h,a /h : U = a A tot U a, U a = U Π a
Decomposition of the propagator Let ψ h be a normalized eigenstate of H h,a with eigenvalue z(h) (assuming Re z = E + o(1), Im z = O(h)). To prove the gap, our aim is to bound from above U N ψ h = e Nt 0 Im z(h)/h and we will need to go up to N M log h, M >> 1. To do this, we use the above quantum partition of unity: U N ψ h = a i A tot, 1 i N U an U an 1 U a2 U a1 ψ h ψ h is microlocalized in H 1 (E) = U a ψ h = O(h ) if any of the a i = E. wavepackets are absorbed in W = U = O(h ). from the escape properties of W a, a A 2, we have U bn... U b1 U a = O(h ) or U a U bn... U b1 = O(h ) if n N 0. As a result, the above sum is dominated by a such that a i A 1 for N 0 < i N N 0.
Using hyperbolicity The following bound (valid for h small enough) relies on the hyperbolicity of Φ t : n = O( log h ), a A n 1, U an U a1 h d/2 (1 + ɛ) nt 0 n j=1 w t0 (W aj ). The last product behaves as e nt0(d 1) λ/2. Due to the prefactor h d/2, the LHS starts to decay exponentially only for n d 1 d (Ehrenfest time). log h λ Take N M log h and sum over all contributions: ( ) N { ( U N ψ h C h d/2 (1+ɛ) Nt 0 d )} w t0 (W a ) C exp Nt 0 +P E (1/2)+2ɛ. 2Mt 0 a A 1 Choosing M d/(2ɛt 0 ), we finally get Im z(h)/h P E (1/2) + 3ɛ.
Proof of the hyperbolic bound The bound on U a is similar as the one proven by [Anantharaman 06] in the case of the geodesic flow on Anosov manifolds. t 0 Λ η Φ W a2 t 0 Λ η Φ( ) Γ ρ +0 ρ I Γ ρ W a1 t 0 Φ( Wa1 ) For any normalized Ψ L 2, the state Π a1 Ψ can be decomposed as Π a1 Ψ = h d/2 I dη f(η) e η + O(h ), where f L2 C and each e η is a normalized WKB state supported on the Lagrangian Λ η (close to Γ +0 ). U e η is also Lagrangian, supported on Φ t 0(Λ η ). After the truncation by Π a2, one has Π a2 U e η J t + 0 (ρ) 1/2 w t0 (W a1 ).
By iterating this procedure, we get U an U a2 U e η n 1 j=1 w t0 (W aj ) + O(h ). Since the RHS is independent of η, we may integrate these norms over η I: n 1 U an U a1 Ψ h d/2 f L1 j=1 w t0 (W aj ), and finally use f L1 I f L2 C.
Final remarks the same procedure can be used for actual resonances of H h. One needs to complexdilate the operator, to obtain a nonselfadjoint H h,θ with properties similar with H h,a. can one show that the semiclassical gap g P (1/2) + ε? Need to control the relative phases of the components U a ψ h. proof of the fractal Weyl law asymptotics? better study semiclassical measures associated with resonant states a nice experimental model is provided by open quantum maps